Only the left-most selection slot in Ultimate Spellbook has any weighting to it. All three other slots are fully random*.
Even for the left-most slot, the weighting that is there only looks to remove ultimates from the pool that would be highly unusable. e.g. We avoid giving long range champions Camille R. Or to put it a different, Ultimate Spellbook tries to ensure that at least your left most slot is usable. And, again, there is no weighting on all three other slots.
* Only exception: Kled can't be offered Mordekaiser R. Which is technically a kind of weighting. This isn't intended, just until we fix the interactions between how Morde R gives bonus health and how Kled uses it. We try to fix these restrictions whenever possible, but haven't been able to get to this one quite yet. e.g. we used to not offer Viktor ult to: Lulu, Shaco, Annie, Leblanc, Ivern until we fixed the underlying reason release.
Why left-slot:
In addition to the idea that "every selection you should have 1 ultimate that's at least vaguely usable", the reason the left-most slot specifically is special is we also default selection to it. If you fail to hit confirm on your ult, you will be defaulted "the last thing you clicked", and the last thing you clicked will default to (intentionally so) the left most slot if you managed to never click the UI at all. e.g. If your adorable cat who you failed to give enough treats to decided to rightfully enact their revenge by ripping out your ethernet cable during the loading screen and you don't get re-connected until after the 30s mark, you should at least have something that's vaguely usable.
We also closely track slot selection rates (from players who confirm their selections) to make sure the left-most slot is not always the implied dominant choice. For visibility, it currently has a 34.9% chance of being selected compared to every other slots ~21.7%
Cho'gath Ult:
Cho ult (and a couple other ults) don't exactly have reduced appearance rate. They're removed from left-slot weighting entirely. Though this reduces the chance of it showing in any one game slightly, it also means that it will show up on all characters more evenly. Compared to the first release's first patch of UltBook, which preferenced Cho ult towards very close range (again, left most slot only here for that preference), Cho ults shows up on non-melee more often than it did before. This is, in large part, due to the rule that only 1 of each ult can show up on a team. We may also flip this eventually to instead be Cho gets preferred for every champion, but haven't yet gotten to digging into the implications of that change enough.
Incidentally, Zoe R is also set to "no-preference" (and a couple others awkward to use ults).